TL;DR: Six weeks old is too early for self-soothing.
Don’t start too young. Parents frequently ask at what age they should start a progressive-waiting approach. It is difficult to give a precise answer to this question, but there are some guidelines. Most children start to sleep through the night on their own within three or four months after birth. (Note that all figures here are based on the due date, so they will be later for children born prematurely.) Newborns do not sleep through the night, and you should not try to make them. Sleep (particularly REM sleep) in the newborn is broken by many wakings. At these times, and during the periods of change from one sleep state to another, an infant’s sleep patterns are particularly vulnerable to disruption. Your newborn should not have to negotiate these periods without comforting and help. At any rate, a full night is too long for a newborn to go between feedings.
By the time a child is around three months old, significant developmental changes have taken place. Now most sleep should be occurring at night, and the pattern of sleep stage cycling should be fairly mature (see Chapter 2). A baby’s sleep patterns often improve markedly around this time, so unless you are having unusually severe problems, it is a good idea to wait until your child is three or four months old before you institute major changes. If at that point you only need to get up briefly once or twice a night to settle your child, you might want to wait a few weeks and see what happens; often, such sleep patterns continue to improve on their own.
But even a two-month-old should not be up more than two or three times during the night. If your baby is up more than that and always goes right back to sleep when you help, you might see if he starts to do better if you do not always respond immediately. (Remember, the methods in this chapter are designed to help you treat habits, not other sleep problems.) If his sleep does not improve over a few days, you should probably give him the benefit of the doubt and wait a few weeks before trying again. (Similarly, as discussed in Chapter 6, you should not have to feed a healthy two-month-old hourly across the night, and it is fine to start spacing out the feedings then, but it is too early to aim at eliminating them altogether.)
If a marked problem persists at three or four months, you can then consider a real effort to make changes. By the time the baby is five months old, you can consider making changes even if his sleep problems are relatively minor.
Ferber, Richard. (2006) "Solve Your Child's Sleep Problems: New, Revised, and Expanded Edition" New York, NY: Fireside: https://www.amazon.com/dp/0743201639/